Single-Photon Atomic Cooling - Raizen Lab - The University of ...

about F, their projections along B (the quantization axis) changes in time.
Therefore the quantum numbers associated with these projections Jz and Iz
(assuming that B is along ˆz) are not good quantum numbers. In contrast,
the projection of F along B is constant in time and so Fz is a good quantum
number. Therefore the Zeeman Hamiltonian (Eq. 2.20) can be written in a
more useful form, in this case, by taking the projection of J and I along F
〈
HZE = µB gJ
F · J〉F · B 〈
+ gI
F(F + 1) F ·
I〉F · B
, (2.21)
F(F + 1)
where the terms in the angled brackets refer to an expectation value. This can
be written more compactly as
where B has been taken along ˆz and gF is given by
gF = gJ
HZE = µBgFFzB, (2.22)
〈 F · J〉 〈
+ gI
F(F + 1) F · I〉
. (2.23)
F(F + 1)
The expectation values can be written in a more illuminating manner; for
example 〈 F · J〉 can be rewritten using the following relation
Squaring both sides of this equation and using
reveals
F + J = I + 2 J. (2.24)
I · J = 1
2 (F 2 − I 2 − J 2 ) (2.25)
〈 F · J〉 = F 2 + J2 − I2 . (2.26)
2
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